Neural Markov chain Monte Carlo: Bayesian inversion via normalizing flows and variational autoencoders

2026-02-23

AI Summary

This work addresses Bayesian inverse problems where the likelihood function is intractable or unavailable and requires repeated evaluations. To tackle this challenge, the authors propose a data-driven inference framework that embeds a variational autoencoder (VAE) within the Markov chain Monte Carlo (MCMC) sampling process to reduce dimensionality and explicitly quantify uncertainty arising from model–reality mismatch. A parameter-conditional normalizing flow is introduced to flexibly model complex observation likelihoods, thereby mitigating overconfident posterior estimates. The method jointly samples latent variables and target parameters, enhanced by a differential evolution Metropolis strategy for improved sampling efficiency. Numerical experiments on railway bridge health monitoring and Darcy flow conductivity field inversion demonstrate the approach’s robustness and effectiveness in handling model discrepancy while delivering reliable uncertainty quantification.

Technology Category

• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper introduces a Bayesian framework that combines Markov chain Monte Carlo (MCMC) sampling, dimensionality reduction, and neural density estimation to efficiently handle inverse problems that (i) must be solved multiple times, and (ii) are characterized by intractable or unavailable likelihood functions. The posterior probability distribution over quantities of interest is estimated via differential evolution Metropolis sampling, empowered by learnable mappings. First, a variational autoencoder performs probabilistic feature extraction from observational data. The resulting latent structure inherently quantifies uncertainty, capturing deviations between the actual data-generating process and the training data distribution. At each step of the MCMC random walk, the algorithm jointly samples from the data-informed latent distribution and the space of parameters to be inferred. These samples are fed into a neural likelihood estimator based on normalizing flows, specifically real-valued non-volume preserving transformations. The scaling and translation functions of the affine coupling layers are modeled by neural networks conditioned on the unknown parameters, allowing the representation of arbitrary observation likelihoods. The proposed methodology is validated on two case studies: (i) structural health monitoring of a railway bridge for damage detection, localization, and quantification, and (ii) estimation of the conductivity field in a steady-state Darcy's groundwater flow problem. The results demonstrate the efficiency of the inference strategy, while ensuring that model-reality mismatches do not yield overconfident, yet inaccurate, estimates.

Problem

• Bayesian inversion
• intractable likelihood
• inverse problems
• model-reality mismatch
• uncertainty quantification

Innovation

• neural Markov chain Monte Carlo
• normalizing flows
• variational autoencoder
• Bayesian inversion
• likelihood-free inference